Ed Miliband has acknowledged that his relationship with brother David has still not fully recovered, more than three years after they fought one another for the Labour leadership.

In an appearance on BBC Radio 4's Desert Island Discs, the Labour leader - who beat his elder brother by the narrowest of margins in the 2010 contest - was asked if his relationship with former foreign secretary David was now "healed". "Healing," he replied.

Despite reports of tensions in the shadow cabinet with Ed Balls, Mr Miliband said that he and the shadow chancellor were determined not to repeat the "dysfunctional" relationship of Tony Blair and Gordon Brown, saying: "Ed Balls and I have seen that movie and we are not going back to it, honestly."

He said he did not feel compromised by the fact that he had won the leadership on the back of union support and insisted that he had "acted really thoroughly to uphold the integrity of the party" in the row with Unite over the selection of a candidate in Falkirk.

Asked what he meant when he said he was bringing socialism back, Mr Miliband said: "Democratic socialism is on our party card. For me it's about a society where there is fairness and justice and greater equality. That's what brought me into the Labour Party.

"We live in a capitalist society. My dad (Marxist academic Ralph Miliband) thought you could abolish capitalism. I don't, but I think it throws up fundamental injustices. What motivates me as a politician is I see injustice and I seek to do something about it."

Mr Miliband said he had no regrets about any of the "big decisions" in his life, and insisted that David was still his best friend.

But he told presenter Kirsty Young: "It has been incredibly tough - really, really tough. I didn't take this decision lightly. I knew it would have an impact on my family and on him.

"What I thought was 'This is going to be hard, but I have got things to say and things to do which I am in a better place than him or others to say and do about the way we need to move on from New Labour and things that need to happen in the country'."

Mr Miliband said that he had "not really" spoken to David - long seen as the heir apparent to Gordon Brown - about the leadership before he announced his decision to stand.

"We had conversations, but they were probably quite elliptical conversations," he said. "We probably danced around it a bit because neither of us was desperate to confront it, I suppose. We didn't have a total heart-to-heart about it."

Once he had decided to stand, he said that their mother Marion "never said to me 'don't do it'" and remained "absolutely scrupulously neutral" during the contest, in which she did not vote. Mr Miliband said it was "hard and difficult" for his mother, but "she wasn't going to take sides in this between her two boys".

Mr Miliband admitted that he probably under-estimated the impact his decision to challenge David would have on his family, and said he could "understand" voters who accused him of putting party loyalties ahead of family loyalties.

But he said he still speaks regularly to David, who refused to serve under him in the shadow cabinet and quit Parliament in April to become president and chief executive of the International Rescue Committee (IRC) in New York.

"We discussed him serving in the shadow cabinet or not on and off over a period, and in the end he said 'I want to lead an organisation, that's what I feel I need the freedom to do in my own way'," said Mr Miliband. "And he is leading the IRC and he seems incredibly exhilarated to be doing it, and I'm very pleased about that."

Family, rather than politics, dominated Mr Miliband's interview on the iconic radio show - which already invited David Cameron to be a castaway in 2006 and Nick Clegg in 2010.

Describing how he tried to fit school runs, nursery drop-offs and bedtime stories with sons Daniel, 4, and Samuel, 3, into his schedule, he said: "My family mean everything to me, it's the most important thing in my life."

Wife Justine is "an amazing person and I feel incredibly blessed and lucky to be married to her", he said. "From the first time I met her I knew there was something very special about her." He did, however, admit that she sometimes jokes that she would have an easier, but less interesting, life if she had married someone else.

One of his eight discs - Angels by Robbie Williams - was chosen to remind him of watching the Live8 concert with Justine "when we were falling in love".

Challenged over his decision not to marry until they had two children and he was leader of the opposition, Mr Miliband said: "We always said we would get married at some point... We did it in what you might call a 21st century way."

Mr Miliband revealed that he did not have any "serious" girlfriends until after leaving Oxford University aged 22 in 1992, admitting: "I was pretty square." He joked that his lack of success might be partly due to the "extremely bad pair of white trousers and purple jumper" he wore to school discos and his "really embarrassing" dancing to A-Ha's Take On Me - another of his desert island choices - saying: "No wonder I didn't pull."

The Labour leader also talked about the inspiration he took from his Jewish parents' flight from the Nazis and the death of many relatives in the concentration camps. He described Marion, now 78, as "the most stoical... go-getting, amazing person" and recalled Ralph as a "great father" and a "lodestar" who had always encouraged him to join in discussions on politics at their north London home.

But Mr Miliband - whose culinary tastes were reflected in his choice of a regular chicken tikka curry takeaway as his desert island luxury - admitted his dad was a "hopeless" cook, who once served him cold spaghetti for two and a half months before he complained.

A self-declared atheist, Mr Miliband revealed that the only time in his life he prayed was before his father's death in 1994: "I'm not a religious person, but I remember saying 'If there's a God, please don't let this happen'."

Choosing Paul Robeson's rendition of left-wing anthem The Ballad of Joe Hill in memory of his father, Mr Miliband recalled: "Both David and I went into politics partly because our parents sort of welcomed us into that conversation and didn't push us out."

Despite his efforts to distance himself from New Labour since taking the leadership, Mr Miliband said he never considered resigning ministerial office over his qualms about issues like 90-day pre-trial detention. Looking ahead to the next election, he insisted there have been no discussions with Liberal Democrats about a possible coalition, saying: "I've got one focus in mind, which is to get a Labour majority."

His choice of book to take to the mythical desert island was Douglas Adams' sci-fi comedy classic The Hitchhiker's Guide to the Galaxy.